Mathematical Methods of Operations Research, Sep 23, 2006
Multichoice games, as well as many other recent attempts to generalize the notion of classical co... more Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.
Central European Journal of Economic Modelling and Econometrics, 2011
Voting power methodology offers insights to understand coalition building in collective decision ... more Voting power methodology offers insights to understand coalition building in collective decision making. This paper proposes a new measure of voting power inspired from Banzhaf (1965) accounting for the proximity between voters by capturing how often they appear in winning coalitions together. Using this proximity index, we introduce a notion of relative linkages among coalition participants as determinant of coalition building. We propose an application to the governance structure of the International Monetary Fund, with linkages being represented by bilateral volumes of trade between voters. The results are able to explain several important features of the functioning of this particular voting body, and may be useful for other applications in international politics.
Voting power methodology offers insights to understand coalition building in collective decision ... more Voting power methodology offers insights to understand coalition building in collective decision making. This paper proposes a new measure of voting power inspired from Banzhaf (1965) accounting for the proximity between voters by cap-turing how often they appear in winning ...
Maître de conférences à l'EHESS Examinateur Ces années de thèse auront été pour moi l'occasion d'... more Maître de conférences à l'EHESS Examinateur Ces années de thèse auront été pour moi l'occasion d'expériences variées dans le monde de la recherche. Pour toutes, sans exception, Michel Grabisch aura été un prodigieux initiateur, présentant conjointement de nombreuses et rares qualités tant sur le plan humain que sur le plan professionnel. Il m'a témoigné une grande patience, m'a soutenu et guidé tout au long de ces années avec une réelle bienveillance. Sa créativité, sa rigueur scientifique, sa détermination et ses talents pédagogiques comptent sans aucun doute pour une grande part dans ma formation et dans mon goût de la recherche. Ces quelques mots sont bien peu de choses en regard de ces quelques lignes qui veulent exprimer ma profonde reconnaissance. Je remercie grandement les professeurs Ulrich Faigle et Stef Tijs de m'avoir fait l'honneur de rapporter ce travail, et d'y avoir investi une grande partie de leur temps. Je remercie vivement Christophe Labreuche, Bruno Leclerc, Jean-François Laslier et Joseph Abdou d'avoir accepté d'examiner cette thèse. Cette thèse n'aurait pu être entreprise si dans un premier temps, le laboratoire Marin Mersenne et l'Ecole docorale de philosophie de l'université Paris 1 ne m'avaient pas chaleureusement accueilli les deux premières années. Je remercie ainsi particulièrement Jean-Bernard Baillon et Denis Pennequin pour leur accueil bienveillant et amical en cette première phase de thèse. Un grand merci également aux membres du SAMOS. J'ai également eu la joie, les deux années suivantes, d'avoir pu rejoindre l'équipe du CERMSEM. Merci à son directeur Bernard Cornet, de m'avoir accueilli, ainsi qu'aux membres enseignants et chercheurs d'avoir contribué à de bonnes conditions de travail en son sein. Un non moindre merci à toute une bande de thésards qui ont su amener joie et bonne humeur au quotidien. Chacun d'eux (elles), avec son originalité, a grandement contribué à une vie de laboratoire des plus sympathiques. Cette expérience n'aurait sans doute pas eu lieu, si je n'avais eu la chance cinq ans plus tôt de suivre la formation du DEA MIASH, avec une fantastique équipe pédagogique à son actif :
A voting situation is given by a set of voters and the rules of legislation that determine minima... more A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley-Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.
The field of cooperative game theory has been enriched these recent years by many new kinds of ga... more The field of cooperative game theory has been enriched these recent years by many new kinds of game, trying to model in a more accurate way the behaviour of players in a real situation. In the classical view of cooperative games, to each coalition of players taking part into the ...
The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a gener... more The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a generalization of capacities (or fuzzy measures) in the context of decision making.
Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of deci... more Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able tocapture awide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index.
Abstract: We propose a generalization of capacities which encompass in a large extent the class o... more Abstract: We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.
Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of deci... more Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bicapacities is studied with new axiomatizations of the interaction index.
Abstract. We propose a new axiomatization of the Shapley value for cooperative games, where symme... more Abstract. We propose a new axiomatization of the Shapley value for cooperative games, where symmetry and efficiency can be discarded and replaced with new natural axioms. From any game, an excluded-player game is built by discarding all coalitions that contain a fixed player. Then it is shown that the Shapley value is the unique value satisfying the linearity axiom, the nullity axiom, the excluded-null-player axiom, and the equity axiom. In the second part, by generalizing the above material, the Shapley value for multichoice games is worked out. Key words. Shapley value, multichoice games, equity, generalized nullity axiom. 1
Abstract: The Shapley value is a central notion defining a rational way to share the total worth ... more Abstract: The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff's laws in the circuit associated to the Hasse diagram of feasible coalitions.
The paper proposes a general approach of interaction between players or attributes. It generalize... more The paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and their interaction indices.
Set functions appear as a useful tool in many areas of decision making and operations research, a... more Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.
Multichoice games, as well as many other recent attempts to generalize the notion of classical co... more Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.
Abstract. In cooperative game theory, the Shapley value is a central notion defining a rational w... more Abstract. In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which...
En théorie des jeux coopératifs, la valeur de Shapley est une notion centrale permettant de défin... more En théorie des jeux coopératifs, la valeur de Shapley est une notion centrale permettant de définir d'une manière rationnelle le moyen de partager la valeur de la grande coalition entre tous les joueurs. Dans le cadre général de ce papier, l'ensemble des coalitions faisables où est défini le jeu forme un ensemble ordonné (par l'inclusion) dont toutes les chaînes maximales ont la même longueur. Nous montrons d'abord que certaines définitions et axiomatisations précédemmentétudiées par Faigle et Kern de la valeur de Shapley restent valables. Notre principale contribution est de proposer une nouvelle axiomatisation quiévite l'axiome de force hiérarchique de Faigle et Kern (difficilement interprétable), considèrant un nouveau moyen de généraliser l'axiome d'anonymat entre les joueurs. Des idées de la théorie des réseauxélectriques sont ensuite empruntées, où nous montrons que notre axiome d'anonymat (regularity axiom) ainsi que l'axiome bien connu d'efficacité (efficiency axiom) correspondent en fait aux deux lois de Kirchhoff d'un circuitélectrique résistif (les noeudsétant données par les coalitions faisables et les branches par les couples de coalitions se précédant). Plus précisément, des analogies sont données entre l'axiome d'efficacité et la loi des noeuds et entre l'axiome d'anonymat et la loi des mailles. Nousétablissons enfin une forme plus faible de l'axiome de monotonie qui est satisfait par la valeur proposée.
The field of cooperative game theory has been enriched these recent years by many new kinds of ga... more The field of cooperative game theory has been enriched these recent years by many new kinds of game, trying to model in a more accurate way the behaviour of players in a real situation. In the classical view of cooperative games, to each coalition of players taking part into the game, an asset or a power (voting games) is associated, and participation is assumed to be of a binary nature, i.e., either a player participates or he does not. From this point, many variations have been introduced, let us cite games with precedence constraints among players (Faigle and Kern [6]) where not all coalitions are valid, ternary voting games (Felsenthal and Machover [7]) where abstention is permitted, bi-cooperative games (Bilbao [2]) where each player can choose to play either in favor, against, or not to play, multichoice games (Hsiao and Raghavan [14]) where each player has a set of m possible ordered actions, fuzzy games (Butnariu and Klement [4], Tijs et al. [18]) which can be seen as a cont...
The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a gener... more The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a generalization of capacities (or fuzzy measures) in the context of decision making. Specifically, let us consider a set X of alternatives in a multicriteria decision making problem, where each alternative is described by a set of n real valued scores (a1, . . . , an). Suppose one wants to compute a global score of this alternative by the Choquet integral w.r.t. a capacity μ, namely Cμ(a1, . . . , an). Then it is well known that the correspondence between the capacity and the Choquet integral is μ(A) = Cμ(1A, 0Ac), ∀A ⊆ N , where (1A, 0Ac) is an alternative having 1 as score on all criteria in A, and 0 otherwise. Such an alternative is called a binary alternative, and the above result says that the capacity represents the overall score of all binary alternatives.
We propose a generalization of capacities which encompass in a large extent the class of Choquet’... more We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.
Mathematical Methods of Operations Research, Sep 23, 2006
Multichoice games, as well as many other recent attempts to generalize the notion of classical co... more Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.
Central European Journal of Economic Modelling and Econometrics, 2011
Voting power methodology offers insights to understand coalition building in collective decision ... more Voting power methodology offers insights to understand coalition building in collective decision making. This paper proposes a new measure of voting power inspired from Banzhaf (1965) accounting for the proximity between voters by capturing how often they appear in winning coalitions together. Using this proximity index, we introduce a notion of relative linkages among coalition participants as determinant of coalition building. We propose an application to the governance structure of the International Monetary Fund, with linkages being represented by bilateral volumes of trade between voters. The results are able to explain several important features of the functioning of this particular voting body, and may be useful for other applications in international politics.
Voting power methodology offers insights to understand coalition building in collective decision ... more Voting power methodology offers insights to understand coalition building in collective decision making. This paper proposes a new measure of voting power inspired from Banzhaf (1965) accounting for the proximity between voters by cap-turing how often they appear in winning ...
Maître de conférences à l'EHESS Examinateur Ces années de thèse auront été pour moi l'occasion d'... more Maître de conférences à l'EHESS Examinateur Ces années de thèse auront été pour moi l'occasion d'expériences variées dans le monde de la recherche. Pour toutes, sans exception, Michel Grabisch aura été un prodigieux initiateur, présentant conjointement de nombreuses et rares qualités tant sur le plan humain que sur le plan professionnel. Il m'a témoigné une grande patience, m'a soutenu et guidé tout au long de ces années avec une réelle bienveillance. Sa créativité, sa rigueur scientifique, sa détermination et ses talents pédagogiques comptent sans aucun doute pour une grande part dans ma formation et dans mon goût de la recherche. Ces quelques mots sont bien peu de choses en regard de ces quelques lignes qui veulent exprimer ma profonde reconnaissance. Je remercie grandement les professeurs Ulrich Faigle et Stef Tijs de m'avoir fait l'honneur de rapporter ce travail, et d'y avoir investi une grande partie de leur temps. Je remercie vivement Christophe Labreuche, Bruno Leclerc, Jean-François Laslier et Joseph Abdou d'avoir accepté d'examiner cette thèse. Cette thèse n'aurait pu être entreprise si dans un premier temps, le laboratoire Marin Mersenne et l'Ecole docorale de philosophie de l'université Paris 1 ne m'avaient pas chaleureusement accueilli les deux premières années. Je remercie ainsi particulièrement Jean-Bernard Baillon et Denis Pennequin pour leur accueil bienveillant et amical en cette première phase de thèse. Un grand merci également aux membres du SAMOS. J'ai également eu la joie, les deux années suivantes, d'avoir pu rejoindre l'équipe du CERMSEM. Merci à son directeur Bernard Cornet, de m'avoir accueilli, ainsi qu'aux membres enseignants et chercheurs d'avoir contribué à de bonnes conditions de travail en son sein. Un non moindre merci à toute une bande de thésards qui ont su amener joie et bonne humeur au quotidien. Chacun d'eux (elles), avec son originalité, a grandement contribué à une vie de laboratoire des plus sympathiques. Cette expérience n'aurait sans doute pas eu lieu, si je n'avais eu la chance cinq ans plus tôt de suivre la formation du DEA MIASH, avec une fantastique équipe pédagogique à son actif :
A voting situation is given by a set of voters and the rules of legislation that determine minima... more A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley-Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley-Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.
The field of cooperative game theory has been enriched these recent years by many new kinds of ga... more The field of cooperative game theory has been enriched these recent years by many new kinds of game, trying to model in a more accurate way the behaviour of players in a real situation. In the classical view of cooperative games, to each coalition of players taking part into the ...
The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a gener... more The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a generalization of capacities (or fuzzy measures) in the context of decision making.
Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of deci... more Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able tocapture awide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index.
Abstract: We propose a generalization of capacities which encompass in a large extent the class o... more Abstract: We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.
Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of deci... more Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bicapacities is studied with new axiomatizations of the interaction index.
Abstract. We propose a new axiomatization of the Shapley value for cooperative games, where symme... more Abstract. We propose a new axiomatization of the Shapley value for cooperative games, where symmetry and efficiency can be discarded and replaced with new natural axioms. From any game, an excluded-player game is built by discarding all coalitions that contain a fixed player. Then it is shown that the Shapley value is the unique value satisfying the linearity axiom, the nullity axiom, the excluded-null-player axiom, and the equity axiom. In the second part, by generalizing the above material, the Shapley value for multichoice games is worked out. Key words. Shapley value, multichoice games, equity, generalized nullity axiom. 1
Abstract: The Shapley value is a central notion defining a rational way to share the total worth ... more Abstract: The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff's laws in the circuit associated to the Hasse diagram of feasible coalitions.
The paper proposes a general approach of interaction between players or attributes. It generalize... more The paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and their interaction indices.
Set functions appear as a useful tool in many areas of decision making and operations research, a... more Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.
Multichoice games, as well as many other recent attempts to generalize the notion of classical co... more Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.
Abstract. In cooperative game theory, the Shapley value is a central notion defining a rational w... more Abstract. In cooperative game theory, the Shapley value is a central notion defining a rational way to share the total worth of a game among players. In this paper, we address a general framework leading to applications to games with communication graphs, where the set of feasible coalitions forms a poset where all maximal chains have the same length. We first show that previous definitions and axiomatizations of the Shapley value proprosed by Faigle and Kern, and Bilbao and Edelman still work. Our main contribution is then to propose a new axiomatization avoiding the hierarchical strength axiom of Faigle and Kern, and considering a new way to define the symmetry among players. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the classical efficiency axiom correspond actually to the two Kirchhoff's laws in the resistor circuit associated to the Hasse diagram of feasible coalitions. We finally work out a weak form of the monotonicity axiom which...
En théorie des jeux coopératifs, la valeur de Shapley est une notion centrale permettant de défin... more En théorie des jeux coopératifs, la valeur de Shapley est une notion centrale permettant de définir d'une manière rationnelle le moyen de partager la valeur de la grande coalition entre tous les joueurs. Dans le cadre général de ce papier, l'ensemble des coalitions faisables où est défini le jeu forme un ensemble ordonné (par l'inclusion) dont toutes les chaînes maximales ont la même longueur. Nous montrons d'abord que certaines définitions et axiomatisations précédemmentétudiées par Faigle et Kern de la valeur de Shapley restent valables. Notre principale contribution est de proposer une nouvelle axiomatisation quiévite l'axiome de force hiérarchique de Faigle et Kern (difficilement interprétable), considèrant un nouveau moyen de généraliser l'axiome d'anonymat entre les joueurs. Des idées de la théorie des réseauxélectriques sont ensuite empruntées, où nous montrons que notre axiome d'anonymat (regularity axiom) ainsi que l'axiome bien connu d'efficacité (efficiency axiom) correspondent en fait aux deux lois de Kirchhoff d'un circuitélectrique résistif (les noeudsétant données par les coalitions faisables et les branches par les couples de coalitions se précédant). Plus précisément, des analogies sont données entre l'axiome d'efficacité et la loi des noeuds et entre l'axiome d'anonymat et la loi des mailles. Nousétablissons enfin une forme plus faible de l'axiome de monotonie qui est satisfait par la valeur proposée.
The field of cooperative game theory has been enriched these recent years by many new kinds of ga... more The field of cooperative game theory has been enriched these recent years by many new kinds of game, trying to model in a more accurate way the behaviour of players in a real situation. In the classical view of cooperative games, to each coalition of players taking part into the game, an asset or a power (voting games) is associated, and participation is assumed to be of a binary nature, i.e., either a player participates or he does not. From this point, many variations have been introduced, let us cite games with precedence constraints among players (Faigle and Kern [6]) where not all coalitions are valid, ternary voting games (Felsenthal and Machover [7]) where abstention is permitted, bi-cooperative games (Bilbao [2]) where each player can choose to play either in favor, against, or not to play, multichoice games (Hsiao and Raghavan [14]) where each player has a set of m possible ordered actions, fuzzy games (Butnariu and Klement [4], Tijs et al. [18]) which can be seen as a cont...
The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a gener... more The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a generalization of capacities (or fuzzy measures) in the context of decision making. Specifically, let us consider a set X of alternatives in a multicriteria decision making problem, where each alternative is described by a set of n real valued scores (a1, . . . , an). Suppose one wants to compute a global score of this alternative by the Choquet integral w.r.t. a capacity μ, namely Cμ(a1, . . . , an). Then it is well known that the correspondence between the capacity and the Choquet integral is μ(A) = Cμ(1A, 0Ac), ∀A ⊆ N , where (1A, 0Ac) is an alternative having 1 as score on all criteria in A, and 0 otherwise. Such an alternative is called a binary alternative, and the above result says that the capacity represents the overall score of all binary alternatives.
We propose a generalization of capacities which encompass in a large extent the class of Choquet’... more We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.
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